1. The problem statement, all variables and given/known data
Find the magnetic field due to a curved wire segment.

2. Relevant equations
Biot-Savart Law (differential form)

dB=$\frac{\mu_{o}i}{4\pi}$ $\frac{d\vec{S}\times \hat{r}}{r^{2}}$

3. The attempt at a solution

In class we found the magnetic field at a point in space (point P) caused by the current running through a wire. Point P is equidistant from every point on the wire call this distance R. dS is a differential element that points along the wire and r hat points toward point P. The angle between dS and r hat is 90 degrees at all points.

The point P is essentially the center of a circle and the wire outlines the edge of a circle.

B=$\frac{\mu_{o}i}{4\pi}$∫$\frac{|d\vec{S}|\times |\hat{r}|}{r^{2}}$

B=$\frac{\mu_{o}i}{4\pi}$∫$\frac{|d\vec{S}||\hat{r}|sinθ}{r^{2}}$

B=$\frac{\mu_{o}i}{4\pi R^{2}}$∫$dS(1)sin90^{o}$

B=$\frac{\mu_{o}i S}{4\pi R^{2}}$ S=R$θ_{1}$
where $θ_{1}$ is the angle swept out between one end of the wire and point P and the other end the wire and point P.

B=$\frac{\mu_{o}i R θ_{1} }{4\pi R^{2}}$

B=$\frac{\mu_{o}i θ_{1} }{4\pi R}$

OK punchline.

I was thinking how could I find the magnetic field at point P due to a piece of wire that outlines a portion of an ellipse. This would mean that the angle between dS and r hat would be a different angle at every point on the wire and the distance between point P and the wire would be different at every point along the wire.

So the integral would be

B=$\frac{\mu_{o}i}{4\pi}$∫$\frac{dS sinθ}{r^{2}}$ where S,θ, and r are all variables. How would I integrate this?

dS=drdθ where dθ the angle between dS and r hat and dr is the infinitesimal change of the radius as the integral adds from one end of the wire to the other end.

B=$\frac{\mu_{o}i}{4\pi}$∫$\frac{drdθ sinθ}{r^{2}}$

This is as far as I can get. Any help would be appreciated. Thanks.

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